Thanks for your reply, Chris.
I went to the IDLE in the ArcGIS 10.1/Python 2.7 folder in the Start menu, and the paths to the python.exe and ILDE were referencing the 64-bit version, so I changed both paths. When I open the IDLE from there, the Shell is now using the 32-bit Python.
Now, I don't know if this makes a difference, but I'm developing my tool through a new Python toolbox, and this is where I'm running into the issue. In the Geoprocessing menu, I had already set the script tool editor to use the 32-bit IDLE before I ran into this error. Even though that path is pointing to the 32-bit version of the IDLE, I still see 64-bit referenced in the Shell when I edit the toolbox and try to run the tool. I tried creating a new Python toolbox, and even the default code doesn't run and I get the same error.
If I read the stack trace right, it looks like when the tool tries to import the arcpy module from the Desktop program files, it fails, which made me assume that the 64-bit Python was causing a problem.
Any more thoughts?